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Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules.  A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${\Q}(n)$ in degrees less than or equal  to $12.$

Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$